The Obama administration opened a last-minute loophole in its health insurance overhaul, offering one-year hardship exemptions from tax penalties to Americans whose private coverage was cancelled because it did not meet the new law's benefit requirements.

Opposition Republicans have fought to rescind the Affordable Care Act, widely known as Obamacare, since it was signed in 2010. They seized on the exemption announced Thursday as further proof that the overhaul is fatally flawed and must be taken off the books.

"President Obama is now admitting, after years of deliberate lies, that ObamaCare is unaffordable and a hardship for millions of Americans," Republican National Committee Chairman Reince Priebus said Friday. "Unilateral delays by this president can't change the basic fact we all know: Obamacare is fundamentally flawed and must be repealed and replaced with a plan that actually improves health care."

The Oct. 1 launch of the HealthCare.gov website became an embarrassment for the White House after problems with the online gateway to froze out millions of potential customers. But the biggest political damage to the president has come from cancellations issued to at least 4 million people whose old plans did not pass muster under the new , which requires more robust benefits.

The insurance industry immediately and loudly complained that the decree, three days before the deadline to sign up for coverage with federal or state insurance marketplaces, would upset the delicate financing balance of the law.

The health overhaul is designed to give all people access to affordable policies. Insurance companies had been enticed into participating in the insurance marketplace by the law's insistence that all Americans buy coverage. The premise was that young and healthy people who often forego coverage would be brought into the pool to offset the cost of paying the claims of older and sicker policy holders.

"This latest rule change could cause significant instability in the marketplace and lead to further confusion and disruption for consumers," said Robert Zirkelbach, spokesman for America's Health Insurance Plans. Only Wednesday, the industry had announced its own accommodation—giving consumers an extra 10 days to pay January's premiums.

Insurance coverage under the new law cannot be denied to people who have pre-existing medical conditions, nor can it be cancelled for policy holders if they become seriously ill. Both tactics were widespread in the insurance industry before the new law went into force.

The loophole, outlined Thursday by Health and Human Services Secretary Kathleen Sebelius, would allow those with cancelled private coverage a one-year grace period to continue with the kind of bare-bones or catastrophic policies the health care overhaul was designed to eradicate.

For Americans wanting uninterrupted coverage on Jan. 1, Monday is the deadline for signing up with the federal insurance marketplace or one of the state markets. The latest decree also would exempt those who claim they can't afford the coverage available in the mandated marketplaces.

The administration was thought to have made the move Thursday in response to complaints from a group of Democratic senators—many of whom face tough re-election campaigns next year—about the private insurance cancellations.

The bulletin announcing the exemption was issued as well in the face of heavy criticism from Republicans and those who had their privately purchased cancelled despite President Barack Obama's repeated promise that any American satisfied with current coverage would be allowed to keep it. While the number of those caught in the surprise cancellations is a small percentage of the U.S. population, those affected still are in the millions. The vast majority of Americans have through their workplace.

On Thursday, the administration estimated at less than 500,000 those who have not yet found new coverage after their old one was canceled.

Obama was roundly criticized for reneging on a longstanding promise that if you liked your plan, you would be able to keep it under his law. The president apologized, and then said insurers could extend those plans for one more year. Most state regulators followed Obama's lead and gave the additional latitude, but it's unclear whether the problem has been fully resolved.

Although the website is now working more smoothly, there's still a concern that technology problems may prevent some people who got cancellations from signing up for a new plan. The industry says it will accept payment of the first month's premiums until Jan. 10. Timely payment is required for the new plan to take effect.

"There still may be a small number of consumers who are not able to renew their existing plans and are having difficulty finding an acceptable replacement," Sebelius wrote Democratic Sen. Mark Warner and several of his colleagues, adding: "These consumers should qualify for this temporary hardship exemption."